Positivity of the two-particle Hamiltonian on a lattice

We consider a two-particle Hamiltonian on the $d$-dimensional lattice $\mathbb Z^d$. We find a sufficient condition for the positivity of a family of operators $h(k)$ appearing after the "separation of the center of mass" of a system of two particles depending on the values of the total quasimomentum $k\in T^d$ (where $T^d$ is a $d$-dimensional torus). We use the obtained result to show that the operator $h(k)$ has positive eigenvalues for nonzero $k\in T^d$.